cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A009574 Expansion of e.g.f. sinh(log(1+x))*exp(x).

Original entry on oeis.org

0, 1, 1, 3, -2, 25, -129, 931, -7412, 66753, -667475, 7342291, -88107414, 1145396473, -16035550517, 240533257875, -3848532125864, 65425046139841, -1177650830516967, 22375365779822563, -447507315596451050, 9397653627525472281, -206748379805560389929
Offset: 0

Views

Author

R. H. Hardin

Keywords

Links

Crossrefs

Cf. A000166, A000240, A054516, A103519.

Programs

  • Magma
    [0] cat [(&+[(k+2)*(-1)^(n-k+1)/Factorial(k): k in [0..n-1]])*( Factorial(n)/2): n in [1..30]]; // G. C. Greubel, Jan 21 2018
  • Maple
    seq(n*(1-(-1)^n*A000166(n-1))/2,n=0..20); # Peter Luschny, Dec 30 2016
  • Mathematica
    CoefficientList[Series[(E^x*x*(2 + x))/(2*(1 + x)), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 23 2015 *)
With[{nn=20},CoefficientList[Series[Sinh[Log[1+x]]*Exp[x],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Mar 23 2015 *)
Table[(-1)^n*n*((-1)^n-Subfactorial[n-1])/2,{n,0,20}] (* Peter Luschny, Dec 30 2016 *)
  • Maxima
    a(n):=n!/2*sum((k+2)*(-1)^(n-k+1)/k!,k,0,n-1); /* Vladimir Kruchinin, Dec 30 2016 */
  • PARI
    x='x+O('x^30); concat([0], Vec(serlaplace(sinh(log(1+x))*exp(x)))) \\ G. C. Greubel, Jan 21 2018
  • Sage
    def A009574():
    a, n = 0, 0
    while True:
        yield a//2
        n += 1
        a = n*(n+1-a)
a = A009574(); [next(a) for  in (0..20)] # _Peter Luschny, Dec 30 2016

Formula

a(n) ~ n! * (-1)^(n+1) / (2*exp(1)). - Vaclav Kotesovec, Jan 23 2015
a(n) = n!/2*Sum_{k=0..n-1}(k+2)*(-1)^(n-k+1)/k!. - Vladimir Kruchinin, Dec 30 2016
a(n) = n*(1-(-1)^n*SF(n-1))/2, where SF(n) is the subfactorial A000166. - Peter Luschny, Dec 30 2016
From Seiichi Manyama, Dec 31 2023: (Start)
a(0) = 0; a(n) = -n*a(n-1) + binomial(n+1,2).
E.g.f.: x * (1+x/2) * exp(x) / (1+x). (End)

Extensions

Extended with signs by Olivier Gérard, Mar 15 1997
First Mathematica program replaced by Harvey P. Dale, Mar 23 2015

A368574 a(n) = n! * Sum_{k=0..n} binomial(k+2,3) / k!.

Original entry on oeis.org

0, 1, 6, 28, 132, 695, 4226, 29666, 237448, 2137197, 21372190, 235094376, 2821132876, 36674727843, 513446190362, 7701692856110, 123227085698576, 2094860456876761, 37707488223782838, 716442276251875252, 14328845525037506580, 300905756025787639951, 6619926632567328080946
Offset: 0

Views

Author

Seiichi Manyama, Dec 31 2023

Keywords

Crossrefs

Cf. A007526, A103519, A368575, A368576.
Cf. A000292, A337001.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); concat(0, Vec(serlaplace(x*sum(k=0, 2, binomial(2, k)*x^k/(k+1)!)*exp(x)/(1-x))))

Formula

a(0) = 0; a(n) = n*a(n-1) + binomial(n+2,3).
E.g.f.: x * (1+x+x^2/6) * exp(x) / (1-x).

A368575 a(n) = n! * Sum_{k=0..n} binomial(k+3,4) / k!.

Original entry on oeis.org

0, 1, 7, 36, 179, 965, 5916, 41622, 333306, 3000249, 30003205, 330036256, 3960436437, 51485675501, 720799459394, 10811991893970, 172991870307396, 2940861795230577, 52935512314156371, 1005774733968978364, 20115494679379576135, 422425388266971109461
Offset: 0

Views

Author

Seiichi Manyama, Dec 31 2023

Keywords

Crossrefs

Cf. A007526, A103519, A368574, A368576.
Cf. A000332, A337002.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); concat(0, Vec(serlaplace(x*sum(k=0, 3, binomial(3, k)*x^k/(k+1)!)*exp(x)/(1-x))))

Formula

a(0) = 0; a(n) = n*a(n-1) + binomial(n+3,4).
E.g.f.: x * (1+3*x/2+x^2/2+x^3/24) * exp(x) / (1-x).

A368576 a(n) = n! * Sum_{k=0..n} binomial(k+4,5) / k!.

Original entry on oeis.org

0, 1, 8, 45, 236, 1306, 8088, 57078, 457416, 4118031, 41182312, 453008435, 5436105588, 70669378832, 989371312216, 14840569694868, 237449115133392, 4036634957288013, 72659429231210568, 1380529155393034441, 27610583107860731324, 579822245265075410934
Offset: 0

Views

Author

Seiichi Manyama, Dec 31 2023

Keywords

Crossrefs

Cf. A007526, A103519, A368574, A368575.
Cf. A000389.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); concat(0, Vec(serlaplace(x*sum(k=0, 4, binomial(4, k)*x^k/(k+1)!)*exp(x)/(1-x))))

Formula

a(0) = 0; a(n) = n*a(n-1) + binomial(n+4,5).
E.g.f.: x * (1+2*x+x^2+x^3/6+x^4/120) * exp(x) / (1-x).

A368762 a(n) = n! * (1 + Sum_{k=0..n} binomial(k+1,2) / k!).

Original entry on oeis.org

1, 2, 7, 27, 118, 605, 3651, 25585, 204716, 1842489, 18424945, 202674461, 2432093610, 31617217021, 442641038399, 6639615576105, 106233849217816, 1805975436703025, 32507557860654621, 617643599352437989, 12352871987048759990, 259410311728023960021
Offset: 0

Views

Author

Seiichi Manyama, Jan 04 2024

Keywords

Crossrefs

Cf. A033540, A368763, A368764.
Cf. A103519, A368766.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace((1+x*sum(k=0, 1, binomial(1, k)*x^k/(k+1)!)*exp(x))/(1-x)))

Formula

a(0) = 1; a(n) = n*a(n-1) + binomial(n+1,2).
a(n) = n! + A103519(n).
E.g.f.: (1 + x * (1+x/2) * exp(x)) / (1-x).

A103520 a(1) = 1, a(n) = product of n successive numbers starting with a(n-1) + 1.

Original entry on oeis.org

1, 6, 504, 65813284680, 1234718152224069224489356305213161520251951242625962440
Offset: 1

Views

Author

Amarnath Murthy, Feb 10 2005

Keywords

Comments

a(6) has 325 digits and a(7) has 2272 digits. - Harvey P. Dale, Aug 23 2014

Examples

			a(2) = 2*3 = 6, a(3) = 7*8*9 = 504.

Crossrefs

Cf. A103519.

Programs

  • Mathematica
    nxt[{n_,a_}]:={n+1,Times@@Range[a+1,a+1+n]}; Transpose[NestList[nxt,{1,1},5]][[2]] (* Harvey P. Dale, Aug 23 2014 *)

Formula

a(n+1) = !{a(n)+n +1}/!{a(n)}

Extensions

More terms from Olaf Voß, Feb 26 2005. (The next term is too large to include.)
Showing 1-6 of 6 results.

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